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Flat knot 6.248

Min(phi) over symmetries of the knot is: [-3,-1,2,2,0,3,3,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.248']
Arrow polynomial of the knot is: -8*K1**2 - 6*K1*K2 + 3*K1 + 4*K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.248', '6.533', '6.1091']
Outer characteristic polynomial of the knot is: t^5+39t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.248']
2-strand cable arrow polynomial of the knot is: -768*K1**6 - 448*K1**4*K2**2 + 2016*K1**4*K2 - 5120*K1**4 + 320*K1**3*K2*K3 - 160*K1**3*K3 - 4288*K1**2*K2**2 - 128*K1**2*K2*K4 + 8040*K1**2*K2 - 1760*K1**2*K3**2 - 128*K1**2*K3*K5 - 512*K1**2*K4**2 - 32*K1**2*K4*K6 - 3592*K1**2 - 640*K1*K2**2*K3 - 384*K1*K2*K3*K4 + 6416*K1*K2*K3 + 3120*K1*K3*K4 + 800*K1*K4*K5 + 40*K1*K5*K6 - 448*K2**4 - 544*K2**2*K3**2 - 216*K2**2*K4**2 + 1384*K2**2*K4 - 4274*K2**2 + 808*K2*K3*K5 + 168*K2*K4*K6 + 8*K3**2*K6 - 2648*K3**2 - 1472*K4**2 - 416*K5**2 - 46*K6**2 + 4902
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.248']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4454', 'vk6.4549', 'vk6.5836', 'vk6.5963', 'vk6.7890', 'vk6.8004', 'vk6.9315', 'vk6.9434', 'vk6.13405', 'vk6.13500', 'vk6.13689', 'vk6.14060', 'vk6.15033', 'vk6.15153', 'vk6.17802', 'vk6.17835', 'vk6.18850', 'vk6.19443', 'vk6.19736', 'vk6.24345', 'vk6.25443', 'vk6.25476', 'vk6.26615', 'vk6.33247', 'vk6.33306', 'vk6.37569', 'vk6.44892', 'vk6.48641', 'vk6.50537', 'vk6.53655', 'vk6.55819', 'vk6.65483']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4O5U4O6U2U6U5U1U3
R3 orbit {'O1O2O3O4O5U4O6U2U6U5U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U5U1U6U4O6U2
Gauss code of K* O1O2O3O4O5U4U1U5U6U3O6U2
Gauss code of -K* O1O2O3O4O5U4O6U3U6U1U5U2
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -3 2 -1 2 1],[ 1 0 -2 2 -1 2 1],[ 3 2 0 3 0 3 1],[-2 -2 -3 0 -1 1 0],[ 1 1 0 1 0 1 0],[-2 -2 -3 -1 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 2 -1 -3],[-2 0 1 -1 -3],[-2 -1 0 -1 -3],[ 1 1 1 0 0],[ 3 3 3 0 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-2,1,3,-1,1,3,1,3,0]
Phi over symmetry [-3,-1,2,2,0,3,3,1,1,-1]
Phi of -K [-3,-1,2,2,2,2,2,2,2,-1]
Phi of K* [-2,-2,1,3,-1,2,2,2,2,2]
Phi of -K* [-3,-1,2,2,0,3,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^4+21t^2
Outer characteristic polynomial t^5+39t^3+10t
Flat arrow polynomial -8*K1**2 - 6*K1*K2 + 3*K1 + 4*K2 + 3*K3 + 5
2-strand cable arrow polynomial -768*K1**6 - 448*K1**4*K2**2 + 2016*K1**4*K2 - 5120*K1**4 + 320*K1**3*K2*K3 - 160*K1**3*K3 - 4288*K1**2*K2**2 - 128*K1**2*K2*K4 + 8040*K1**2*K2 - 1760*K1**2*K3**2 - 128*K1**2*K3*K5 - 512*K1**2*K4**2 - 32*K1**2*K4*K6 - 3592*K1**2 - 640*K1*K2**2*K3 - 384*K1*K2*K3*K4 + 6416*K1*K2*K3 + 3120*K1*K3*K4 + 800*K1*K4*K5 + 40*K1*K5*K6 - 448*K2**4 - 544*K2**2*K3**2 - 216*K2**2*K4**2 + 1384*K2**2*K4 - 4274*K2**2 + 808*K2*K3*K5 + 168*K2*K4*K6 + 8*K3**2*K6 - 2648*K3**2 - 1472*K4**2 - 416*K5**2 - 46*K6**2 + 4902
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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